Pin a Rigid Body Object
A rigid body constraint always needs two rigid bodies. Add a passive rigid body above the cone: Cube.



Select the empty and make it a rigid body hinge constraint, with the new Cube and Cone as the objects.

Other tweaks:


The hinge constraint rotates around its Z axis. You will have to reorient it.
Move the origin of the cone to its geometric center.
Set the Cones shape to Cone.